Non-linear wave propagation solutions by Fourier transform perturbation

Abstract A Fourier transform perturbation method is developed and used to obtain uniformly valid asymptotic approximations of the solution of a class of one-dimensional second order wave equations with small non-linearities. Multiple time scales are used and the initial-value problem on the infinite line is solved by Fourier transforming the wave equation and expanding the Fourier transform in powers of the small parameter. The non-linearity involves only the first partial derivatives of the dependent variable and the determination of the leading approximation is reduced to the solution of a pair of coupled non-linear ordinary differential equations in Fourier space. Examples are given involving a convolution non-linearity and a Van-der-Pol non-linearity.